Abstract. In this paper we provide various properties of Rad-â-supplemented modules. In particular, we prove that a projective module M is Rad-.
DOI: 10.2478/auom-2013-0015 An. S ¸ t. Univ. Ovidius Constant ¸a
RAD-
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Vol. 21(1), 2013, 225–238
-SUPPLEMENTED MODULES Erg¨ ul T¨ urkmen
Abstract In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P ∗ ) if and only if R is an artinian serial ring and J 2 = 0, where J is the Jacobson radical of R. Finally, we show that every Rad-supplemented module is Rad-⊕-supplemented over dedekind domains.
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Introduction
Throughout this paper, it is assumed that R is an associative ring with identity and all modules are unital left R-modules. A submodule N of an R-module M will be denoted by N ≤ M . A submodule L ≤ M is said to be essential in M , denoted as L E M , if L ∩ N ̸= 0 for every nonzero submodule N ≤ M . Dually, a submodule N of M is called small (in M ) and denoted by N
